Theorem 2.22 ([1]}) Let \(M\in \operatorname{mod}(R)\)  and let \(N\in \operatorname{Mod}(R)\) . Assume \(\operatorname{\textnormal {CI-dim}}_R(M)<\infty \) . Then,
\(\widehat{\operatorname{Tor}}_i^R(M,N)=0 \ \text{ for all } i\in \mathbb {Z}\Longleftrightarrow \widehat{\operatorname{Tor}}_i^R(M,N)=0\ \text{ for all } i\ll 0 \Longleftrightarrow \operatorname{Tor}_i^R(M,N)=0 \ \text{ for all } i\gg 0.\) 


